6.1 Sliding surfaces and reaching conditions
3 min readโขaugust 14, 2024
Sliding surfaces are key to sliding mode control, defining desired system behavior. They're designed to make trajectories converge and slide along them, leading to stability and . Reaching conditions ensure trajectories hit the surface fast and stay there.
These concepts are crucial for making sliding mode control work. By understanding sliding surfaces and reaching conditions, you'll grasp how to design controllers that force systems to behave as desired, even with uncertainties or disturbances.
Sliding Surfaces in Control
Definition and Role
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- Sliding surfaces are hypersurfaces in the state space that represent the desired dynamic behavior of the system
- Designed such that system trajectories converge to and slide along the surface, leading to the desired system response (stability, tracking, robustness)
- Defined by a linear combination of system states, with coefficients chosen to achieve desired closed-loop dynamics
- Reduces the order of system dynamics and enforces performance specifications
Design Considerations
- Design depends on control objectives and system dynamics
- For linear systems, is designed by placing closed-loop poles at desired locations in the complex plane
- Ensures stability and performance
- Coefficients determined using pole placement techniques (Ackermann's formula, bass-gura method)
- For nonlinear systems, sliding surface is designed based on desired closed-loop dynamics (linear or nonlinear)
- Should consider the relative degree of the system, which determines the order of the sliding mode controller
- Minimize reaching phase duration and reduce chattering by incorporating reaching law parameters or using higher-order sliding mode control techniques
Reaching Conditions for Sliding Mode
Definition and Purpose
- Ensure system trajectories converge to the sliding surface in finite time and remain on the surface thereafter
- Most common is the ฮท-
- Requires Lyapunov function of sliding variable to decrease at a rate proportional to its magnitude
- Constant rate reaching law:
- is the sliding variable
- is a positive constant
- is the signum function
Alternative Reaching Laws
- Constant plus proportional rate reaching law
- Power rate reaching law
- Exponential reaching law
- Offer different convergence properties and chattering reduction compared to constant rate reaching law
Stability Analysis of Sliding Mode
Lyapunov Stability Theory
- Used to analyze stability of sliding mode control systems
- Lyapunov function is defined, which is positive definite and radially unbounded
- Time derivative is negative definite along system trajectories
Stability of Sliding Motion
- For sliding mode control, Lyapunov function is typically chosen as a quadratic function of the sliding variable:
- Stability of sliding motion is guaranteed if time derivative of Lyapunov function is negative definite
- Ensures system trajectories converge to and remain on the sliding surface
- Reachability condition, combined with stability of sliding motion, ensures overall stability of sliding mode control system
Sliding Surface Design for Control Objectives
Linear Systems
- Sliding surface designed by placing closed-loop poles at desired locations in the complex plane
- Ensures stability and performance
- Coefficients determined using pole placement techniques
- Ackermann's formula
- Bass-gura method
Nonlinear Systems
- Sliding surface designed based on desired closed-loop dynamics (linear or nonlinear)
- Should consider the relative degree of the system
- Determines the order of the sliding mode controller
- Minimize reaching phase duration and reduce chattering
- Incorporate reaching law parameters
- Use higher-order sliding mode control techniques
Key Terms to Review (16)
Automotive systems: Automotive systems refer to the various components and technologies within vehicles that work together to ensure performance, safety, and efficiency. These systems include engine control, braking, steering, and stability control, often utilizing nonlinear control techniques to enhance functionality under varying driving conditions.
Boundary Layer Approach: The boundary layer approach is a concept used in control theory to analyze systems that exhibit switching behaviors, particularly in sliding mode control. This technique focuses on the regions where the system's dynamics change abruptly, allowing for the design of robust controllers that can handle uncertainties and disturbances effectively. By examining the boundary layers, engineers can establish conditions under which the system reaches a desired sliding surface and maintains stability.
Chattering Phenomenon: The chattering phenomenon refers to the rapid oscillations that occur when a control system attempts to maintain a sliding mode on the sliding surface. This behavior is often undesirable as it can lead to excessive wear on mechanical components and instabilities in the system. It typically arises due to high-frequency switching of the control input, which can occur when the system is close to the sliding surface but not quite on it, causing it to repeatedly overshoot and undershoot the target state.
Exponential Convergence: Exponential convergence refers to the rapid approach of a system's state towards a desired equilibrium point or sliding surface over time, characterized by a convergence rate that decreases exponentially as time progresses. This behavior is crucial in control systems as it indicates the efficiency and speed with which the system can eliminate errors and settle into stability. Systems exhibiting exponential convergence demonstrate not only quick stabilization but also robust performance in response to disturbances.
Lyapunov stability: Lyapunov stability refers to the property of a dynamic system where, if it is perturbed from its equilibrium position, it will eventually return to that position over time. This concept is essential in assessing how systems respond to disturbances and is foundational in the design and analysis of control systems, especially nonlinear ones.
Nonlinear dynamic systems: Nonlinear dynamic systems are systems in which the output is not directly proportional to the input, leading to complex behavior that can change over time. These systems can exhibit phenomena like chaos, bifurcations, and limit cycles, making their analysis and control more challenging compared to linear systems. Understanding these systems is crucial for designing control strategies, such as sliding mode control, that can handle uncertainties and maintain performance despite the inherent complexities.
Reachability Condition: The reachability condition refers to the ability of a system to reach a specific state from any initial state within a finite amount of time, provided the appropriate control inputs are applied. This concept is vital in the context of sliding mode control, as it ensures that a system can transition to a desired sliding surface where desirable dynamic properties, such as stability and robustness, can be achieved.
Reaching Condition: The reaching condition is a critical requirement in sliding mode control that determines when a system trajectory enters a predefined sliding surface. This condition ensures that the system states will converge to the desired sliding surface, where robust control is achieved despite uncertainties and disturbances. A successful reaching condition guarantees that once the system reaches this surface, it will stay there, maintaining stability and performance.
Robotics: Robotics is the interdisciplinary branch of engineering and science focused on the design, construction, operation, and use of robots. These automated machines can perform tasks typically done by humans, often in complex or hazardous environments, making them essential in various applications from manufacturing to medical assistance.
Robustness: Robustness refers to the ability of a system to maintain performance and stability despite uncertainties, disturbances, or variations in its parameters. This quality is essential in control systems, as it ensures that the system can adapt to changes in the environment or internal dynamics without significant degradation in performance.
Sliding Mode Observer: A sliding mode observer is a type of state observer used in control systems that combines the robustness of sliding mode control with the estimation capabilities of observers. It enables the reconstruction of the system's states in the presence of uncertainties and disturbances by forcing the estimation error to slide along a predefined surface. This technique enhances the reliability of state estimation, particularly in systems with significant noise or model inaccuracies.
Sliding Mode Reaching Law: The sliding mode reaching law is a control strategy used in sliding mode control systems to ensure that the system state reaches a predefined sliding surface in a finite time. This law plays a crucial role in the dynamics of the system, ensuring that once the state is on the sliding surface, it behaves according to desired performance specifications. The reaching law defines how the control input is applied to steer the system state towards the sliding surface while compensating for uncertainties and disturbances.
Sliding surface: A sliding surface is a defined manifold in the state space of a dynamical system where the system's behavior becomes invariant to disturbances and uncertainties, allowing for robust control. This concept is crucial in control strategies that involve sliding mode control, which utilizes the sliding surface to drive the system's trajectory to a desired state and maintain it there despite external influences. Understanding the characteristics of the sliding surface is essential for achieving performance and stability in various applications, including observers and robotic systems.
System robustness: System robustness refers to the ability of a control system to maintain performance and stability in the presence of uncertainties, disturbances, or variations in system parameters. It emphasizes the resilience of the system against unexpected changes, ensuring that it can still achieve desired outcomes despite potential challenges. Robustness is crucial for the reliability of sliding mode control, as it dictates how well the system can handle deviations and still meet the reaching conditions required for sliding surfaces.
Time-delay systems: Time-delay systems are dynamic systems where there is a significant delay between the input and output responses, often due to the time it takes for signals to travel or processes to occur. These delays can complicate the control strategies used for system regulation, making it challenging to ensure stability and performance. Understanding how to manage these delays is crucial in designing effective controllers, particularly when dealing with sliding surfaces and reaching conditions.
Tracking Error: Tracking error is the difference between the desired output of a control system and the actual output it produces, often represented as the error signal. This discrepancy is crucial for assessing the performance and accuracy of various control strategies, especially in nonlinear systems where maintaining desired performance can be challenging due to inherent system dynamics.